Wicketkeeper Bairstow, 28, said he was targeted during the first Test in Brisbane but did not reveal the nature of the words directed towards him.
He did say they were not related to the ‘headbutt’ incident between him and Cameron Bancroft before the series.
Bairstow also said he felt “stitched up” over how that incident was treated.
“Some other things, apart from the ‘headbutt’ business, were said by Australia in the middle but what they were is staying there,” he wrote in a Daily Mail column.
“The second Test was played in a good spirit – tough but fair. There were some verbals from both teams but this time nothing crossed the line.
“Clashes like we saw in Adelaide are part of sport. I’m not making an issue of it. Only if they are said again would the matter go further.
“We move on. Hopefully it’s gone now. We just need to get on with trying to get back in this series.”
Australia’s players have admitted targeting England with “smarter” sledging during the Ashes series, which they lead 2-0 with the third Test due to start in Perth on Thursday, 14 December (02:30 GMT).
Former England wicketkeeper Matt Prior told BBC Radio 5 live on Monday that the contentious Australian sledging related to an issue which “hasn’t come out for various reasons”, adding some comments had “rightfully” upset some players.
Bairstow’s close friend and former Yorkshire team-mate Ryan Sidebottom also criticised the Australian tactics.
“I’m all for being in the moment and sledging and trying to win for your team, but do it in the right manner, not bring in other things that’s in your personal life or anything like that,” he told BBC Radio 5 live.
“Someone’s said something that’s upset him and I suppose he’s come out and mentioned that he doesn’t really want to hear that kind of thing so well done to him for mentioning it, but it’s not ideal is it?
“You don’t want this kind of thing happening but at least he’s mentioned it so Australia know about it and may nip it in the bud.” (Courtesy BBC)